JMC 2018

Halves them to 45°

HintA square's diagonal is a line of symmetry.

The whyBecause the diagonal is a line of symmetry, it bisects the right angles at both ends, giving 45° each way and making two right-angled isosceles triangles. Angles at a corner shared by a square and another shape are then just a matter of adding.

7; 8; 10; 12

HintGreek number words: hepta, octo, deka, dodeka.

The whyRegular polygon questions almost always need the number of sides for a perimeter or an angle. A regular polygon has equal sides, so its perimeter is the side length times the number of sides.

b − a + 1

HintCount from 3 to 5 on your fingers before you subtract.

The whyFrom 12 to 30 inclusive there are 30 − 12 + 1 = 19 numbers, not 18. Subtracting alone counts the gaps; the +1 adds back the number you started on. Strictly between a and b there are b − a − 1. The same fencepost logic counts terms in a sequence and posts along a fence.

larger

HintOne cake shared between more people.

The whyOne eighth is less than one quarter because the whole is cut into more pieces. It is also why a fraction of a fraction always shrinks: the denominators multiply, so the pieces get smaller. With equal numerators, compare denominators — backwards.

6

HintCount the faces of a dice.

The whyEach face is a × a and there are six of them. For a solid with no hidden pockets, count the faces you can see from each of the six directions — top, bottom and four sides — rather than the faces of each cube.

5

HintOne test looks at the last digit.

The whyBecause 3 and 5 have no common factor, being a multiple of both is the same as being a multiple of 15. So check the last digit (0 or 5) and the digit sum (a multiple of 3). The same works for 45 = 9 × 5 and 12 = 3 × 4, but not for 12 = 2 × 6, because 2 and 6 share a factor.

The leading digit

HintThe most significant place decides the most.

The whyFix the front of the number first, then fill greedily: for the largest, the biggest remaining digits in decreasing order; for the smallest, the smallest remaining in increasing order. For 'closest together', give the two numbers neighbouring front digits, then push one down and the other up.

11

HintSix squares joined edge to edge that fold up with no overlap — there are more than you'd guess.

The whySix of the eleven have a strip of four squares with one square on each side. To test a net quickly, check that no four squares meet at a point (a 2 × 2 block) and that the two 'flap' squares fold onto opposite faces, not the same one.

A flat shape that folds up to make the solid

HintUnfold a cardboard box and lay it on the table.

The whyA net must contain every face once and fold with no overlap. For a cube that means six squares; for a triangular prism, three rectangles and two triangles.

4

HintThink about the worst case: how many picks can go by with no match at all?

The whyOne more than the number of colours forces a pair, because the worst case is one of each colour before the next pick must repeat one. Always reason from the worst case: what is the most you could pick without success?

360° ÷ n

HintAll the exterior angles are equal and together make one full turn.

The whyFor a regular polygon, exterior angle = 360° ÷ n and interior angle = 180° − exterior. Running it backwards is how you find n: an interior angle of 156° means an exterior angle of 24°, so n = 360 ÷ 24 = 15.

All sides equal and all angles equal

HintThe one word that lets you assume everything is identical.

The why'Regular' is the word that unlocks a diagram: it hands you equal sides (so isosceles triangles from the centre) and equal angles (so 360° ÷ n exterior, and 180° minus that interior). Without it, none of those follow.